12x12 Queens Puzzle #105

Difficulty: easy. Place 12 queens on this 12x12 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 33-step reasoning is written out below.

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Step-by-step solution for puzzle #105

What follows is a complete solve of puzzle #105, worked the way a person works it rather than the way a computer does. It runs to 33 deductions — 12 queen placements and 21 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the proof by contradiction.

Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on columns, proof by contradiction, last square in a row.

Show the full walkthrough (28 steps) — contains the solution

  1. Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. Amber is down to 4 candidates in rows 2–4, columns 10 and 12, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.

  3. Step 3. Pale yellow cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 9 squares in all.

  4. Step 4. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 13 squares come off.

  5. Step 5. Pale yellow cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 1 square in all.

  6. Step 6. The red region has 5 squares left, spread across rows 6–8, columns 2–4. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

  7. Step 7. Wherever rose ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.

    Puzzle #105 after step 7: 1 of 12 queens placed
    The board after step 7 — 1 of 12 queens placed.

  8. Step 8. Here is the pairing argument. The amber, orange, sea green, and rose regions have no open squares outside columns 9–12 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 19 squares come off in one stroke.

  9. Step 9. Slate cannot leave column 8 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 6 squares in all.

  10. Step 10. Wherever khaki ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  11. Step 11. Test row 2, column 12 for amber: place a queen there, run the forced consequences through, and column 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  12. Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 12. Follow the forced moves that would create and column 10 runs out of open squares entirely — impossible. So that square is ruled out for good.

  13. Step 13. Test row 4, column 12 for amber: place a queen there, run the forced consequences through, and column 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  14. Step 14. The amber region is down to one open square, at row 2, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 10, the remaining amber squares, and the ring of neighbours around row 2, column 10.

    Puzzle #105 after step 14: 2 of 12 queens placed
    The board after step 14 — 2 of 12 queens placed.

  15. Step 15. Only row 9, column 9 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 9, in orange and in the ring around that square is now dead.

  16. Step 16. All of the rose region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  17. Step 17. Test row 3, column 3 for blue: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.

  18. Step 18. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 5. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.

  19. Step 19. Test row 3, column 6 for blue: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  20. Step 20. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 2. Follow the forced moves that would create and row 11 runs out of open squares entirely — impossible. So that square is ruled out for good.

  21. Step 21. Test row 4, column 3 for blue: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

    Puzzle #105 after step 21: 3 of 12 queens placed
    The board after step 21 — 3 of 12 queens placed.

  22. Step 22. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 4. Follow the forced moves that would create and row 3 runs out of open squares entirely — impossible. So that square is ruled out for good.

  23. Step 23. Test row 4, column 5 for blue: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.

  24. Step 24. Row 4 has a single open square left, at column 6 in the mint region. Every row needs a queen, so this one is forced. Cross out the whole of row 4 and column 6, the remaining mint squares, and the ring of neighbours around row 4, column 6.

  25. Step 25. Look along row 3: one square survives, row 3, column 2, which belongs to blue. A row cannot be left empty, so the queen goes there. Everything else in row 3, in column 2, in blue and in the ring around that square is now dead.

  26. Step 26. Row 5 has a single open square left, at column 8 in the slate region. Every row needs a queen, so this one is forced. Row 5 and column 8 are now spoken for, along with the squares touching that queen.

  27. Step 27. The red region has 4 squares left, spread across rows 6–8, columns 3 and 4. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.

  28. Step 28. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 5; pale yellow has one square left, so its queen takes row 7, column 1; red has one square left, so its queen takes row 6, column 3; sea green has one square left, so its queen takes row 10, column 12; rose has one square left, so its queen takes row 12, column 11; teal has one square left, so its queen takes row 11, column 4. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 11, column 4.

    Puzzle #105 after step 28: 12 of 12 queens placed
    The board after step 28 — 12 of 12 queens placed.

That completes the board: 12 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.

About 12x12 Queens puzzle #105

Puzzle #105 is one of the easy-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #105, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these twelve regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #105 are: indigo (1 square), orange (2 squares), pale yellow (3 squares), rose (4 squares), amber (8 squares), red (9 squares), sea green (11 squares), teal (18 squares), slate (20 squares), blue (21 squares), khaki (23 squares), mint (24 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the mint region is the roomiest area on the grid with 24 squares.

Every color takes one queen and one only, whatever its size — which is why the 23 square difference between indigo and mint is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #105, packing 100% of its 1×1 bounding box, whereas khaki straggles across its 6×10 box at just 38%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Here is the full footprint of every color on puzzle #105, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo: 1 square across rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.

Orange covers 2 squares, spanning 1 row and 2 columns (rows 9–9, columns 9–10). It never leaves row 9, so its queen is locked to that row.

Pale yellow occupies 3 squares within rows 7–9, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Rose: 4 squares across rows 11–12, columns 9–11.

Amber covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 9–12).

Red occupies 9 squares within rows 5–8, columns 1–4.

Sea green: 11 squares across rows 6–12, columns 9–12.

Teal covers 18 squares, spanning 3 rows and 7 columns (rows 10–12, columns 1–7).

Slate occupies 20 squares within rows 2–7, columns 7–12.

Blue: 21 squares across rows 1–5, columns 1–6.

Khaki covers 23 squares, spanning 6 rows and 10 columns (rows 7–12, columns 2–11).

Mint occupies 24 squares within rows 1–8, columns 4–9.

Where to start on this board

Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 7 and 7. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #105.

Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.

Work outward to the other flat regions next. Orange and rose are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.

The tightest lines on puzzle #105

A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 3 colors. Columns 8 and 12 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 7 is the crowded extreme at 6 colors — worth revisiting later, not now.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

The whole middle game reduces to two sweeps. Sweep the twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.

The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.

Why puzzle #105 is rated easy

The rating reflects how much the board concedes up front. Puzzle #105 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 23 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.

Stuck on #105 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.

Techniques that work well at 12x12

At 12x12 you are managing 144 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #105 is row 10.

A non-spoiling distribution check: solved, #105 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
  5. There is always exactly one valid solution. Use logic, never guessing, to find it.

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Questions about this puzzle

How hard is 12x12 Queens puzzle #105?

Puzzle #105 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #105 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #105, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #105?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.